Local Distinguishability Of Orthogonal Product States And Its Application In Cryptography

Because quantum algorithms have great advantages over classical algorithms in speed, people pay more and more attention to the field of quantum information. Local distinguishability of quantum states is one of the most important contents in this field. The so-called local discrimination refers to the persons that each holds a particle of a state identify the state by Local Operations and Classic Communication(LOCC). In this paper, we mainly study local distinguishability of orthogonal product states and its application in some quantum cryptographic algorithms.It is thought that a set of orthogonal product states can be distinguished by local operations and classic communication since there is no entanglement between the particles of a product state. However,Bennett et al. proposed nine orthogonal product states that cannot be distinguished by LOCC in a 3(?)3 quantum system. This shows that entanglement is not a necessary condition for nonlocal property of a set of orthogonal quantum states. Although people have devoted a lot of efforts to the study of orthogonal product states, we know little about local distinguishability of orthogonal product states on high-dimensional systems. We study local distinguishability of both bipartite and multipartite high-dimensional quantum systems in depth. We give a general method to construct an orthogonal product basis that cannot be distinguished by local operations and classic communication both in a bipartite quantum system and in a multipartite quantum system. We discuss the application of local distinguishability of orthogonal product basis in the designs of quantum cryptographic protocols. The main results of this paper are as follows.1. We give a general method of constructing two kinds of orthogonal product basises that cannot be distinguished by local operations and classic communication in arbitrary bipartite high-dimensional quantum system. Of the two kinds of orthogonal product basises, one is uncompletable and the other is completable. They are different from the existing results. In particular, we give so far the smallest number of locally indistinguishable states of a completable orthogonal product basis in arbitrary bipartite high-dimensional quantum system. All the results lead to a better understanding of the structures of locally indistinguishable product basises in arbitrary bipartite quantum systems.2. In order to solve the difficult problem of constructing orthogonal product basis that cannot be locally distinguished, we present a generic method to construct a completable product basis with only 2n members and an uncompletable product basis with a smaller number of members in a multipartite quantum system. Each of the two basises cannot be distinguished by local operations and classical communication. They show quantum nonlocality without entanglement in any multipartite high-dimensional quantum system. All the results lead to a better understanding of the structures of locally indistinguishable product basises in arbitrary multipartite quantum systems.3. We first construct an orthogonal product basis with n parties each holding a (n+1)/2 dimensional system, where n>5 and n is odd, and prove its local indistinguishability by showing that all the positive operator-valued measure (POVM) elements of each party can only be proportional to the identity operator to make further discrimination feasible. Then we prove a subset of this product basis is still locally indistinguishable with a weaker condition. This means that it is a sufficient condition but not a necessary condition for local indistinguishability of a multipartite product basis that all the POVM elements of each party can only be proportional to the identity operator to make further discrimination feasible. Moreover, we exhibit that a small set of n-partite product states, which contains only 2n members, cannot be perfectly distinguished even in any locally extended Hilbert space,where n?5 and n is odd.4. Based on local indistinguishability of orthogonal product states,we design a quantum secret sharing protocol, a quantum key agreement protocol, a quantum signature protocol and a quantum proxy signature protocol, respectively. These protocols have all the properties that correspond to classic cryptographic protocols. They can guarantee the security of the transmitted message since the different particles of a product state are not transmitted at the same time during the transmission process.